We first consider the constraints arising from electroweak precision observables (EWPOs) [1]}, [2]}.
We introduced two scalars to the SM, one of them is a real singlet scalar which can mix with the SM Higgs boson.
Both the SM Higgs boson and the real singlet scalar will contribute to the SM gauge boson self-energies at loop-level and finally induce corrections to the oblique parameters \(S\) , \(T\) , and \(U\) .
The other one is a pseudo-Goldstone boson \(\chi \)  which becomes a DM candidate due to an accidental \(\mathbb {Z}_2\)  symmetry hidden in our \(\mathbb {Z}_3\)  symmetry model. The pseudo-Goldstone boson do not mix with SM particles and thus do not affect the oblique parameters. Constraints from the EWPO's can alter the distribution of \(m_{\chi }\)  via the parameters \(m_{\mathcal {S}}\)  and \(\theta \) .
